Golden Ratio

The Golden Section (Latin: sectio aurea) or also “Divine division” (Latin: proportio divina) consists in the division of a distance $\overline{AB}$ in two sections $\overline{AM}$ and $\overline{MB}$ , so that the length of the longer section $\overline{AM}$ to the length of the shorter section $\overline{MB}$ behaves like the length of the total distance $\overline{AB}$ to the length of the longer section $\overline{AM}$ (cf. Figure 1):

Figure 1: Golden section as distance ratio

Expressed as an equation, this is called

$\lvert\overline{AM}\rvert:\lvert\overline{MB}\rvert=\lvert\overline{AB}\rvert:\lvert\overline{AM}\rvert\quad (\ast).$

This division of a line in the ratio of the golden section has been known for millennia and is still of particular importance today in architecture and art as an aesthetic principle. The first preserved precise description of the golden section is by Euclid (ca. 300 BC). The Franciscan friar Luca Pacioli di Borgo San Sepolcro (1445–1514), who taught mathematics in Perugia, Italy, called this division the “Divine Division“. The first known calculation of the golden ratio as an approximation of 1.6180340 for the ratio $\lvert\overline{AM}\rvert:\lvert\overline{MB}\rvert$ (see Figure 1) was communicated by the Tübingen professor Michael Maestlin to his former student Johannes Kepler (1571–1630) in 1597.

However, the term Golden Section, which is commonly used today, was first used in a mathematics textbook published in 1835 by Martin Ohm (1792–1872).

The golden section plays a significant role in architecture as well as in the visual arts. Thus, many buildings of antiquity (e.g. the front façade of the built around 440 BC Parthenon Temple of the Athens Acropolis), but also famous buildings in later centuries (e.g. the gate hall of the Lorsch Monastery (around 770), the Florence Cathedral (1294 Start of construction), Notre Dame of Paris (1163–1345) and the Old City Hall in Leipzig (1556–1557)) follow the Golden Section in their proportions (in the latter, it is the distances of the entrance under the tower to the two sides of the building).

In addition, numerous sculptures by ancient Greek sculptors are considered evidence of the use of the golden section, as are a number of Renaissance paintings, including those by Raphael, Leonardo da Vinci and Albrecht Dürer (e.g. his self-portrait of 1500 and the copperplate engraving “Melencolia” of 1514).

The golden ratio is also admired time and again because it can be observed in nature, such as in the five-petalled symmetry of bellflowers, the ivy aralia leaf and the starfish. On the human body, too, a number of proportions can be found that are associated with the golden ratio. They were first studied systematically by Adolph Zeisig in the middle of the 19th century. In particular, the fact that on the human body the belly button divides the body size as the golden section has become famous (see exhibit in MATH ADVENTURE LAND).

And now … the mathematics:

If we set $a\coloneqq\lvert\overline{AM}\rvert$ and $b\coloneqq\lvert\overline{MB}\rvert$ , then $\lvert\overline{AB}\rvert=a+b$ . So from the above equation $(\ast)$ it follows $a:b=(a+b):a.$

Now set $\Phi\coloneqq a/b$ . Then, this gives

$\Phi=a/b=(a+b)/a=1+b/a=1+\Phi^{-1}.$

Multiplying this by $\Phi$ gives the quadratic equation

$\Phi^2-\Phi-1=0,$

which has the zeros $\Phi_{1,2}=\frac{1\pm\sqrt{5}}{2}$ . But since $\Phi$ is obviously positive, $\Phi=\frac{1+\sqrt{5}}{2}$ must hold. This value is also often referred to as the golden number.

Literature

[1] Pacioli, L.: Divina Proportione, Venedig, 1509.

[2] Beutelspacher, A., und Petri, B.: Der Goldene Schnitt, Heidelberg, Berlin, Oxford, 1996.

[3] Hemenway, P.: Divine Proportion. Phi in Art, Nature and Science, New York, 2005.